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Inclusive neutrino reactions in 127I
NUCLEAR PHYSICS
A
Nuclear Physics A584 (1995) 665-674
ELSEVIER
Inclusive neutrino reactions in
127I
S.L. M i n t z a, M. P o u r k a v i a n i b a Physics Department, Florida International University, Miami, FL 33199, USA bMassachusetts Institute of Technology, Cambridge, MA 02139, USA
Received 19 July 1994; revised 3 October 1994
Abstract We obtain the cross section for the reaction ve + 127I----) e - + X from threshold to 53 MeV via two different models, both of which make use of data for the total muon-capture rate in 127I. The neutral-current cross section for the reaction v +1271 ~ v + X is also calculated over the same energy range. Average cross sections over the Michel spectrum are also obtained for both of these reactions. The consequences of these rates are discussed with respect to the use of 127I in neutrino detectors.
1. Introduction I n t e r e s t in t h e r e a c t i o n , v e + 1271 --~ e - q - X stems f r o m t h e s u g g e s t i o n [1-3] t h a t 1271 m i g h t b e s u i t a b l e m a t e r i a l for a s o l a r - n e u t r i n o d e t e c t o r . T h e r e a r e a n u m b e r o f r e a s o n s for b e l i e v i n g t h a t this m i g h t b e so. First, 1271 has a g i a n t d i p o l e r e s o n a n c e at only 15 M e V a n d c a n e m i t a n u c l e o n a b o v e 7.2 M e V . I n a d d i t i o n 1271 has a l a r g e n e u t r o n excess w h i c h w o u l d t e n d to drive u p r e a c t i o n rates. F i n a l l y it is easy to c o m b i n e i o d i n e w i t h o t h e r e l e m e n t s such as 23Na o r 7Li to b u i l d d e t e c t o r s . T h e s a m e f e a t u r e s t h a t m i g h t m a k e 127I a u s e f u l s o l a r - n e u t r i n o d e t e c t o r also m a k e it a n a t t r a c t i v e t a r g e t for M i c h e l - s p e c t r u m n e u t r i n o s a v a i l a b l e at L A M P F a n d at K A R M E N . I n a d d i t i o n t h e M i c h e l s p e c t r u m p e a k s at a r o u n d 37 M e V w h i c h is a b o u t 22 M e V a b o v e t h e g i a n t d i p o l e r e s o n a n c e so t h a t c l o s u r e m i g h t b e a n a p p r o p r i a t e a p p r o x i m a t i o n w h i c h w o u l d simplify calculations. I n d e e d , o n e g r o u p is a l r e a d y u n d e r t a k i n g a n 1271 n e u t r i n o e x p e r i m e n t [4] at L o s A l a m o s , a n d such a n e x p e r i m e n t has a l r e a d y b e e n p r o p o s e d for K A R M E N [5]. T h u s a calculat i o n for t h e p r o c e s s e s v e q-127I ~ e - + X a n d v -t-1271 ~ V -q- X o v e r t h e r a n g e o f t h e M i c h e l s p e c t r u m is timely. W e shall o b t a i n cross s e c t i o n s for t h e r e a c t i o n ve q-127I ~ e - q - X , using two m e t h o d s . T h e first m e t h o d [6] m a k e s u s e o f a t e n s o r to d e s c r i b e t h e h a d r o n i c p a r t 0375-9474/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 5 - 9 4 7 4 ( 9 4 ) 0 0 4 9 0 - 0
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of the matrix element squared and a random-phase approximation to limit the tensor to two terms. The second method [7] makes use of a closure approximation and a simple model of the nucleon-current matrix element to obtain a cross section. Both methods make use of the total muon-capture rates in the calculation of the cross section. In addition we shall obtain averaged cross sections over the Michel spectrum for this reaction and for the neutral current reaction, ~ + 127I ~ ~, + X. Thus, in this paper, we shall be primarily interested in the energy range of the Michel spectrum rather than the solar-neutrino spectrum. However, there is some overlap between the two spectra and therefore the results might have some bearing on the usefulness of 127I as a solar-neutrino detector.
2. Matrix elements
We consider first the reaction ve +127I ~ e - + X. The starting point for both methods is the matrix element G Mki = ~ - c o s 0c ~ey~(1 -- ys)u~ < ( k [ J a*(0) 1127I),
(1)
where k is a particular final state and J~*(0) = V~*(0) -A*~(0).
(2)
We shall only outline the process by which we obtain the cross section as the details have already appeared in the literature [6,8]. We simply note that the cross section may be written as
my /d3Pe I Mki ] 2 me °c = ~-~k 2ME~, J Ee(2~) 3 d3pk X 2Elc(27r)3 (27r)4~4(P~ +Pe - P v - P i ) .
(3)
The quantity I Mki 12 is given by
Ia k i I2
G2
c°siOe.L'~a(k]JS(O) ] '27I)(k I Jff(0) 1127I}* . 2m~m e
(4)
The quantity L ~x is the lepton tensor appropriate to this process and is given by
~ . ~ . ,~ =PeP~ --Pe "P~g~* +PffPex -- ~,~13*,~
(5)
In order to work with average quantities, we assume an average nuclear excitation of 6 given by
M.-M~=a,
(6)
S.L. Mintz, M. Pourkaviani /Nuclear Physics A584 (1995) 665-674
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where 6 is clearly a function of the incoming neutrino energy. We also assume on the basis of our knowledge of individual states that the interaction is largely in the forward direction and that (E~> --- E~ - 6
(7)
( P e ) = ~(E~ - 6) 2 - m~2 .
(8)
and
Although the value of 6 will vary with incoming neutrino energy, above the giant dipole resonance at 15 M e V it should increase slowly in our region of interest. This enables us to easily obtain ( E e) and ( P c ) from Eqs. (7) and (8) over a large part of the range of neutrino energy of interest. W e can, using these averages, obtain the quantity
c~ c°s2°c j 2.,"da~ ~ ( k ] J~(O) llZTI)(k l J](O)1127I)*L ¢x °'c
2 ME~,
k
(Ipel) × 2M-2ge+2g
~ cos o e ( g e ) / ( l P e l ) "
(9)
The hadronic part of Eq. (9) may be replaced as ( k I JS(0) 1127I)(klJ](O) 1127I) * = Q;~o_(ei, ( q ) ) ,
(10)
k which is a tensor. W e have previously shown [6,8] that this tensor may be reduced t o the form
Q'~" = ag '~" + ~ P i " P / ' .
(11)
The cross section is then found to be
ere =
G 2 COS20c ( I Pe [ ) ( E e ) D 4rr M(M+E~) '
(12)
where D =/3 - 2 a ,
(13)
and an impulse-approximation-based calculation [7] gives D ( q 2) as:
D = ozo - boq 2.
(14)
We assume this simple q2 dependence for D, which we may also be written as D=ao +bolq2]. Thus our result depends upon two parameters, a 0 and b 0. Ideally, we would m a k e use of muon-capture data and electron-scattering data to determine D exactly. T h e r e is data for m u o n capture in 127I, but a little consideration will make clear that this is not what we want. If we consider the 12C case, m u o n capture leads to 12B states whereas neutrino reactions lead to 12N states. But 12B and 12N arc
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S.L. Mintz, M. Pourkaviani /Nuclear Physics A584 (1995) 665-674
from the same isospin multiplet and have corresponding levels. That is why our technique works well for this case. However, the 127I case is very different. Here there are 21 excess neutrons and we therefore need a nucleus in the same nmltiplet with 21 excess protons. Of course there is no such stable nucleus but we shall make use of the G o u l a r d - P r i m a k o f f formula [9,10] with the most recent values of parameters to create a total muon capture for this nucleus which we shall call aZ7w. The G o u l a r d - P r i m a k o f f formula is remarkably accurate over the entire range of nuclei, predicting most rates to a few percent and so we expect it to yield a reasonable result for our nucleus. The general form of the G o u l a r d - P r i m a k o f f result is given by GPT = ZeffG 4 1 1 q- G2~-~ F~o
--
G3
2Z
a4 ~
+ -
8AZ
'
(15a)
where G 1 = 261, G 2 = - 0 . 0 4 0 , G 3 = - 0 . 2 6 and G 4 = 3.24. On the other hand, total muon-capture results are available for the process / x - + 127I---~ u~ + X. Following an essentially identical calculation [6], we find for the total muon-capture rate for essentially any nucleus
C[ @(0) [2G2 c o s 2 O c ( E v ) 2 O /~TOT =
8,1TMi(M i +
m,)
(15b)
But we must have the two results, Eq. (15a) and Eq. (15b), equal for the same nucleus. We can therefore consider the ratio
GP (]P~OT)W _~_ (F~oPT)w ( G GP TOT)I ( ff'MPT)I
(16a)
where GP stands for the O o u l a r d - P r i m a k o f f result of Eq. (15a) and MP stands for the result of Eq. (15b). In this ratio, because C = ( l e f t / Z ) 4 and I q~(0) [ 2 = 23o/3(1 +m~/M) -3, there is substantial cancellation in the ratio of Eq. (16a). The remaining expressions are readily evaluated yielding Dw
Zw -
DI
-
ZI
× 4.08 = 5.7.
(16b)
The total muon-capture rate for 127I is known [9] and is given by FTOT ~ (11.2 ___ 0.11) × 10 6 s -~. The correction factor, C, has already been given and for 27I yields C = 0.093,
(17)
where we have used [ll] Zeff = 29.27. We note that even a small error in Zeff will have large consequences, but the effective values for Z are believed to be well known. Using E~ = 90.62 MeV, we obtain D = 1.244 × 1012.
(18)
This number carries a 5 to 10 percent error due primarily to the error in FToa-, the error in Zeff and the uncertainty in the average excitation of 1 2 7 I .
S.L. Mintz, M. Pourkaviani / Nuclear Physics A584 (1995) 665-674
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W e have at this point a value for D at q2 appropriate for m u o n capture, but we need an additional piece of information to determine D completely. In the case of 12C we could rely on inclusive electron-scattering data and some impulse-approximation results to fully obtain D. H e r e this is not possible. However, an impulse-approximation result [7] yields a value f o r / 5 = O(qZ)/o(o) given by /) =
1 - [(A-Z)/ZA]6(q 1- [(A-Z)/ZA]8(0)
2) '
(19a)
where
e 2t
k ro ] \ ---q-O-]' with r 0 =
B/A1/3,d/ro
(19b)
= 1.5, and where from Eq. (14) we may write
b = 1 - b°q a=- 1 -b'oq 2.
(19c)
ao
This finally yields a q2 for our case given by /) = 1 - 0.1898q e.
(19d)
W e note that 6(q2) is unrelated to the 3 of Eq. (6), fi(E~), which is the mass difference of excited states as a function of neutrino energy. Above the giant dipole resonance we expect closure to be applicable and so for E~ > 30 M e V we set 6(E,,) = 15 MeV. W e choose this value from previous experience [7] with total muon-capture rates where good results are obtained at 15 to 20 M e V above the giant dipole resonance with closure. F r o m our experience with the t2C case, below 30 M e V we use a decreasing 6 given by
6(E,)
= 1.667 X 1 0 - 2 E 2 - 6.0023 × 10 -3.
(20)
We have tried several different forms for 6(E,), and the precise form of the function does not have m o r e than an effect of a few percent on the value of (o-c}. W e are now able to evaluate Eq. (12). W e next consider a model [7] developed by Kim and Mintz for treating neutrino reactions in nuclei. This is strictly a closure approximation and should begin to be accurate at about 15 to 20 M e V about the giant dipole resonance which is assumed to saturate the matrix element. W e do not include here a full derivation which is already in the literature [7] but simply mention the salient points. As already mentioned, the starting point is Eq. (1). The quantity S+~, is defined by
dG E f~7-(ilQ~+)(q)
k)
(21)
k
with
Q(,+-)(q) - f dxJ+(x, O) e 'x'q.
(22)
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We assume that low-lying states (particularly those associated with the giant dipole resonance) saturate the sum in Eq. (21) so that we may write, applying closure,
S ~+__ - (i[O~+)(q)Q~-)(q) ]i).
(23)
Making use of translational invariance, we can show that the neutrino cross section and the total muon-capture rate depend upon essentially the same matrix element: (i] Q(+)" Q(-) ] i) + ( i l 4t~(+)o(-) [i ) . 0 40 The matrix elements themselves can be evaluated by making use of an impulse approximation for the weak current, namely A
J(~+-)(x, O) = Y'~ (F,)a~(X-ra) ,
(24)
a--1
where
(F~)a = gv6~,4 - (1 - ~ , 4 ) gA(~r,*) ~•
(25)
With these definitions we may write d/2
f -4~ [(iiQ(+).Q(-)li)+ (i[t~(+)t~(-)[i)]40 40 q2)). =Z(g~¢ + 3g 2 ) ( A1 - Z B2( A
(26)
This quantity occurs as mentioned above in the expressions for Fa-oT and for o-c. F r o m Eq. (26) the only difference is the qZ at which the matrix element is evaluated. As we have constructed a value for the quantity Fa-oa-, we may write crc as
2~rFTowEePe 1 - [ ( A - Z ) / 2 A ] 6 ( q 2) ~rc= Ci(aZ)3m~(~ 2) 1 - 3 . 1 3 [ ( A - Z ) / Z A ]
(27)
In Eq. (27), 6(q e) is given by Eq. (19). We know that Eq. (27) is valid only well above closure so we expect it to produce an underestimate for (o-c). We therefore take a worst-case (lowest-estimate) value for ~(q2), namely ~(q2) _~ 3.38 1 + 0.091-25m.
.
(28)
We now have everything necessary to calculate cross sections for the reaction Pe + -127I --> e - + X. Finally we note that from Ref. [6], we may write the cross section for inclusive neutral-current neutrino scattering as GZ(E') 2 ON=
8M2~
× 0.832D.
(29)
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671
25
20
15 c%(10-4%m2) 10
5
1'0
15
20
25 3~0 E (MeV)
Fig. 1. Cross section for the reaction v e -1-127I-> e calculation is via the tensor model.
+X
3'5
4'0
4'5
5'0
55
as a function of neutrino energy. This
W e are t h e r e f o r e able to evaluate the c h a r g e d - c u r r e n t cross sections in b o t h models and to find the n e u t r a l - c u r r e n t cross section.
3. Results I n Fig. 1 we plot the results of the tensor m o d e l for the c h a r g e d - c u r r e n t cross section, o-c. As already noted, at and above 30 M e V we assume 6(E~) is given by the constant, 15 MeV, but below 30 M e V we write it as given in Eq. (20). T h e r e is some arbitrariness i n h e r e n t in this but we know that 6(E~) must on average fall in this region. W e note that (o-c ) is not greatly affected by any reasonable choice of functions. F r o m Fig. 1 and the Michel s p e c t r u m we obtain the s p e c t r u m - a v e r a g e d value (o-c} = 638.52 x 1 0 - 4 2 c m 2,
(30)
w h e r e we expect a 30% error to be associated with this number. I n Fig. 2, we plot the c h a r g e d - c u r r e n t cross section using the second m o d e l described in this paper. This m o d e l yields zero below the giant dipole r e s o n a n c e at 15 M e V , as it must f r o m the assumptions used in its derivation. F r o m this result we obtain a M i c h e l - s p e c t r u m - a v e r a g e d cross section (o-c} = 303.52 X 1 0 - 4 2 c m 2.
(31)
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S.L. Mintz, M. Pourkaviani / N u c l e a r Physics A584 (1995) 665-674
12
10
crc(10-40cm2)6
01•5
20
2~5
3'0
3'5 E (MeV)
Fig. 2. Cross section for the reaction v e + 1 2 7 I - - ~ e - + X calculation is via the closure-approximation model.
4'0
4'5
5'0
55
as a function of neutrino energy. This
Finally, in Fig. 3, we plot the neutral-current results for the tensor model. This result leads to a neutral-current cross section for the reaction u + 127I--* ~, + X, averaged over the Michel spectrum, given by (o"N } = 265.18 × 10 -42 cm 2.
(32)
9 8
7 6
5 O-g( l O-4Ocrn2) 4 3 2 1
00
5
1'0
1'5
2'0
2'5
3'0
3'5
4'0
4~5
5'0
55
E (MeV) Fig. 3. Cross section for the reaction v + 127I ~ ~v-F X as a function of neutrino energy. This calculation is via the tensor model.
S.L. Mintz, M. Pourkaviani/ Nuclear PhysicsA584 (1995) 665-674
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W e have therefore obtained results for the inclusive charged- and the inclusive neutral-current electron-neutrino reactions in 127I. We now discuss these results in m o r e detail.
4. Discussion At present there are no published estimates of the Michel-spectrum-averaged inclusive neutrino cross sections in 127I. However, there are some informal estimates circulating which are based to a large extent on the substantial neutron excess in 127I. These estimates are of the order of 600 × 10 -42 cm 2. These are in line with our calculations as well. It is interesting to compare the small-nucleus case, i.e. 12C, with a large-nucleus case such as la7I. In the small-nucleus case, various models gave a wide range of results in this energy range. But in the large-nucleus case almost all reasonable models give a similar result. This may be due to the fact that the large nuclei with their very closely spaced levels are more like a nuclear soup so that microscopic-structure effects are not very important and a Fermi-gas t r e a t m e n t works very well. On the other hand, for a small nucleus like 12C, the structure is of p a r a m o u n t importance but is difficult to handle in calculations. This makes a semi-phenomenological treatment useful in such cases. For both of the models described here, the results are strongly tied to the total muon-capture rate. This connection is worth discussing. The function D = a 0 + bo [q21 is determined from muon-capture results as well as from additional impulse-approximation results. In the case of ~2C done earlier, b 0 was relatively small. Because I q2[ is of the order of 0.75 m ~ , D was dominated by a o. This remained true for the neutrino reaction, which was assumed to take place in the forward direction so that q2 would be quite small. Furthermore, even though the impulse-approximation value for b 0 was larger than that obtained via electronscattering data both were small enough so that the domination of D by a 0 was not affected. In the case of nuclei with large neutron excesses, this is no longer true, and because of this, a 0 is relatively smaller in these cases. But it is precisely this value for a 0 which will dominate the neutrino reaction. Thus effectively, our second model with its relatively large value for b 0 and its early cutoff at 15 MeV, the assumed center of the giant dipole resonance, yields a minimal value for ( % ) while the tensor model yields, we believe, a more realistic result. Thus we believe that a value for (o-c) of the order of (500 ~ 700) × 10 -42 cm 2 might be expected. For the region below the giant dipole resonance, i.e., the region of interest for solar-neutrino observation, we do not expect our value for o-c to be particularly accurate as we have only an impulse-approximation result to serve as a basis for the calculation. However, if it were possible to obtain inclusive electronscattering data for this (and indeed the entire range of interest), a much better calculation would be possible. Nevertheless, the result in this region, from threshold to 10 MeV, appears to favor the low end of current estimates. We should also r e m a r k that if the convergence of theoretical results is confirmed by experiment for the 127I case, it will demonstrate the ability of the G o u l a r d - P r i m a k o f f result for total muon-capture rates to work under the widest range of circumstances.
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S.L. Mintz, M. Pourkaviani /Nuclear Physics A584 (1995) 665-674
Finally, w e wish to n o t e t h a t s o m e e a r l y results for t h e r e a c t i o n u~ + 12C ~ / x - + X have b e e n r e p o r t e d [12] f r o m Los A l a m o s . T h e m u o n n e u t r i n o s a r e p r o d u c e d f r o m i n - f l i g h t - p i o n decays a n d have an e n e r g y r a n g e f r o m 10 to 260 M e V . T h e s e p r e l i m i n a r y results a r e in g o o d a g r e e m e n t with t h e two m o d e l s d e s c r i b e d h e r e which y i e l d e s s e n t i a l l y t h e s a m e r e s u l t for t h e e n e r g y r a n g e for which t h e r e a c t i o n is p o s s i b l e (since t h e s p e c t r u m is p e a k e d well a b o v e t h e g i a n t d i p o l e r e s o n a n c e ) b u t in s t r o n g d i s a g r e e m e n t with a F e r m i - g a s - m o d e l calculation. This gives us s o m e slight a d d e d c o n f i d e n c e c o n c e r n i n g t h e s e m o d e l s . H o w e v e r , t h e r e is clearly m u c h w o r k b o t h t h e o r e t i c a l l y a n d e x p e r i m e n t a l l y to b e d o n e b e f o r e any u n d e r s t a n d i n g of t h e w e a k h a d r o n i c c u r r e n t in inclusive p r o c e s s e s c a n b e c l a i m e d .
References [1] S.E. Woolsley, D.H. Hartrnann, R.D. Hoffman and W.C. Haxton, Ap. J. 356 (1990) 272. [2] W.C. Haxton, Phys. Rev. Lett. 60 (1988) 768. [3] J. Engel, AIP Conf. Proc. 243, Intersections between particle and nuclear physics 1991, ed. W.T.H. Van Oers (American Institute of Physics, New York, (1992) p. 1134. [4] K. Lande, private communication. [5] J.A. Edgington, private communication. [6] M. Pourkaviani and S.L. Mintz, Nucl. Phys. A 573 (1994) 501. [7] C.W. Kirn and S.L. Mintz, Phys. Rev. C 31 (1985) 274. [8] S.L. Mintz and M. Pourkaviani, J. Phys. G 20 (1994) 925. [9] T. Suzuki, D.F. Measday and J.P. Roalsvig, Phys. Rev. C 35 (1987) 2212. [10] B. Goulard and H. Primakoff, Phys. Rev. C 10 (1974) 2034. [11] K.W. Ford and J.G. Wills, Nucl. Phys. 35 (1962) 295. [12] C. Athanassopoulos, Contributed talk (American Physical Society, Washington, DC, 1994).
A
Nuclear Physics A584 (1995) 665-674
ELSEVIER
Inclusive neutrino reactions in
127I
S.L. M i n t z a, M. P o u r k a v i a n i b a Physics Department, Florida International University, Miami, FL 33199, USA bMassachusetts Institute of Technology, Cambridge, MA 02139, USA
Received 19 July 1994; revised 3 October 1994
Abstract We obtain the cross section for the reaction ve + 127I----) e - + X from threshold to 53 MeV via two different models, both of which make use of data for the total muon-capture rate in 127I. The neutral-current cross section for the reaction v +1271 ~ v + X is also calculated over the same energy range. Average cross sections over the Michel spectrum are also obtained for both of these reactions. The consequences of these rates are discussed with respect to the use of 127I in neutrino detectors.
1. Introduction I n t e r e s t in t h e r e a c t i o n , v e + 1271 --~ e - q - X stems f r o m t h e s u g g e s t i o n [1-3] t h a t 1271 m i g h t b e s u i t a b l e m a t e r i a l for a s o l a r - n e u t r i n o d e t e c t o r . T h e r e a r e a n u m b e r o f r e a s o n s for b e l i e v i n g t h a t this m i g h t b e so. First, 1271 has a g i a n t d i p o l e r e s o n a n c e at only 15 M e V a n d c a n e m i t a n u c l e o n a b o v e 7.2 M e V . I n a d d i t i o n 1271 has a l a r g e n e u t r o n excess w h i c h w o u l d t e n d to drive u p r e a c t i o n rates. F i n a l l y it is easy to c o m b i n e i o d i n e w i t h o t h e r e l e m e n t s such as 23Na o r 7Li to b u i l d d e t e c t o r s . T h e s a m e f e a t u r e s t h a t m i g h t m a k e 127I a u s e f u l s o l a r - n e u t r i n o d e t e c t o r also m a k e it a n a t t r a c t i v e t a r g e t for M i c h e l - s p e c t r u m n e u t r i n o s a v a i l a b l e at L A M P F a n d at K A R M E N . I n a d d i t i o n t h e M i c h e l s p e c t r u m p e a k s at a r o u n d 37 M e V w h i c h is a b o u t 22 M e V a b o v e t h e g i a n t d i p o l e r e s o n a n c e so t h a t c l o s u r e m i g h t b e a n a p p r o p r i a t e a p p r o x i m a t i o n w h i c h w o u l d simplify calculations. I n d e e d , o n e g r o u p is a l r e a d y u n d e r t a k i n g a n 1271 n e u t r i n o e x p e r i m e n t [4] at L o s A l a m o s , a n d such a n e x p e r i m e n t has a l r e a d y b e e n p r o p o s e d for K A R M E N [5]. T h u s a calculat i o n for t h e p r o c e s s e s v e q-127I ~ e - + X a n d v -t-1271 ~ V -q- X o v e r t h e r a n g e o f t h e M i c h e l s p e c t r u m is timely. W e shall o b t a i n cross s e c t i o n s for t h e r e a c t i o n ve q-127I ~ e - q - X , using two m e t h o d s . T h e first m e t h o d [6] m a k e s u s e o f a t e n s o r to d e s c r i b e t h e h a d r o n i c p a r t 0375-9474/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 5 - 9 4 7 4 ( 9 4 ) 0 0 4 9 0 - 0
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of the matrix element squared and a random-phase approximation to limit the tensor to two terms. The second method [7] makes use of a closure approximation and a simple model of the nucleon-current matrix element to obtain a cross section. Both methods make use of the total muon-capture rates in the calculation of the cross section. In addition we shall obtain averaged cross sections over the Michel spectrum for this reaction and for the neutral current reaction, ~ + 127I ~ ~, + X. Thus, in this paper, we shall be primarily interested in the energy range of the Michel spectrum rather than the solar-neutrino spectrum. However, there is some overlap between the two spectra and therefore the results might have some bearing on the usefulness of 127I as a solar-neutrino detector.
2. Matrix elements
We consider first the reaction ve +127I ~ e - + X. The starting point for both methods is the matrix element G Mki = ~ - c o s 0c ~ey~(1 -- ys)u~ < ( k [ J a*(0) 1127I),
(1)
where k is a particular final state and J~*(0) = V~*(0) -A*~(0).
(2)
We shall only outline the process by which we obtain the cross section as the details have already appeared in the literature [6,8]. We simply note that the cross section may be written as
my /d3Pe I Mki ] 2 me °c = ~-~k 2ME~, J Ee(2~) 3 d3pk X 2Elc(27r)3 (27r)4~4(P~ +Pe - P v - P i ) .
(3)
The quantity I Mki 12 is given by
Ia k i I2
G2
c°siOe.L'~a(k]JS(O) ] '27I)(k I Jff(0) 1127I}* . 2m~m e
(4)
The quantity L ~x is the lepton tensor appropriate to this process and is given by
~ . ~ . ,~ =PeP~ --Pe "P~g~* +PffPex -- ~,~13*,~
(5)
In order to work with average quantities, we assume an average nuclear excitation of 6 given by
M.-M~=a,
(6)
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where 6 is clearly a function of the incoming neutrino energy. We also assume on the basis of our knowledge of individual states that the interaction is largely in the forward direction and that (E~> --- E~ - 6
(7)
( P e ) = ~(E~ - 6) 2 - m~2 .
(8)
and
Although the value of 6 will vary with incoming neutrino energy, above the giant dipole resonance at 15 M e V it should increase slowly in our region of interest. This enables us to easily obtain ( E e) and ( P c ) from Eqs. (7) and (8) over a large part of the range of neutrino energy of interest. W e can, using these averages, obtain the quantity
c~ c°s2°c j 2.,"da~ ~ ( k ] J~(O) llZTI)(k l J](O)1127I)*L ¢x °'c
2 ME~,
k
(Ipel) × 2M-2ge+2g
~ cos o e ( g e ) / ( l P e l ) "
(9)
The hadronic part of Eq. (9) may be replaced as ( k I JS(0) 1127I)(klJ](O) 1127I) * = Q;~o_(ei, ( q ) ) ,
(10)
k which is a tensor. W e have previously shown [6,8] that this tensor may be reduced t o the form
Q'~" = ag '~" + ~ P i " P / ' .
(11)
The cross section is then found to be
ere =
G 2 COS20c ( I Pe [ ) ( E e ) D 4rr M(M+E~) '
(12)
where D =/3 - 2 a ,
(13)
and an impulse-approximation-based calculation [7] gives D ( q 2) as:
D = ozo - boq 2.
(14)
We assume this simple q2 dependence for D, which we may also be written as D=ao +bolq2]. Thus our result depends upon two parameters, a 0 and b 0. Ideally, we would m a k e use of muon-capture data and electron-scattering data to determine D exactly. T h e r e is data for m u o n capture in 127I, but a little consideration will make clear that this is not what we want. If we consider the 12C case, m u o n capture leads to 12B states whereas neutrino reactions lead to 12N states. But 12B and 12N arc
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from the same isospin multiplet and have corresponding levels. That is why our technique works well for this case. However, the 127I case is very different. Here there are 21 excess neutrons and we therefore need a nucleus in the same nmltiplet with 21 excess protons. Of course there is no such stable nucleus but we shall make use of the G o u l a r d - P r i m a k o f f formula [9,10] with the most recent values of parameters to create a total muon capture for this nucleus which we shall call aZ7w. The G o u l a r d - P r i m a k o f f formula is remarkably accurate over the entire range of nuclei, predicting most rates to a few percent and so we expect it to yield a reasonable result for our nucleus. The general form of the G o u l a r d - P r i m a k o f f result is given by GPT = ZeffG 4 1 1 q- G2~-~ F~o
--
G3
2Z
a4 ~
+ -
8AZ
'
(15a)
where G 1 = 261, G 2 = - 0 . 0 4 0 , G 3 = - 0 . 2 6 and G 4 = 3.24. On the other hand, total muon-capture results are available for the process / x - + 127I---~ u~ + X. Following an essentially identical calculation [6], we find for the total muon-capture rate for essentially any nucleus
C[ @(0) [2G2 c o s 2 O c ( E v ) 2 O /~TOT =
8,1TMi(M i +
m,)
(15b)
But we must have the two results, Eq. (15a) and Eq. (15b), equal for the same nucleus. We can therefore consider the ratio
GP (]P~OT)W _~_ (F~oPT)w ( G GP TOT)I ( ff'MPT)I
(16a)
where GP stands for the O o u l a r d - P r i m a k o f f result of Eq. (15a) and MP stands for the result of Eq. (15b). In this ratio, because C = ( l e f t / Z ) 4 and I q~(0) [ 2 = 23o/3(1 +m~/M) -3, there is substantial cancellation in the ratio of Eq. (16a). The remaining expressions are readily evaluated yielding Dw
Zw -
DI
-
ZI
× 4.08 = 5.7.
(16b)
The total muon-capture rate for 127I is known [9] and is given by FTOT ~ (11.2 ___ 0.11) × 10 6 s -~. The correction factor, C, has already been given and for 27I yields C = 0.093,
(17)
where we have used [ll] Zeff = 29.27. We note that even a small error in Zeff will have large consequences, but the effective values for Z are believed to be well known. Using E~ = 90.62 MeV, we obtain D = 1.244 × 1012.
(18)
This number carries a 5 to 10 percent error due primarily to the error in FToa-, the error in Zeff and the uncertainty in the average excitation of 1 2 7 I .
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W e have at this point a value for D at q2 appropriate for m u o n capture, but we need an additional piece of information to determine D completely. In the case of 12C we could rely on inclusive electron-scattering data and some impulse-approximation results to fully obtain D. H e r e this is not possible. However, an impulse-approximation result [7] yields a value f o r / 5 = O(qZ)/o(o) given by /) =
1 - [(A-Z)/ZA]6(q 1- [(A-Z)/ZA]8(0)
2) '
(19a)
where
e 2t
k ro ] \ ---q-O-]' with r 0 =
B/A1/3,d/ro
(19b)
= 1.5, and where from Eq. (14) we may write
b = 1 - b°q a=- 1 -b'oq 2.
(19c)
ao
This finally yields a q2 for our case given by /) = 1 - 0.1898q e.
(19d)
W e note that 6(q2) is unrelated to the 3 of Eq. (6), fi(E~), which is the mass difference of excited states as a function of neutrino energy. Above the giant dipole resonance we expect closure to be applicable and so for E~ > 30 M e V we set 6(E,,) = 15 MeV. W e choose this value from previous experience [7] with total muon-capture rates where good results are obtained at 15 to 20 M e V above the giant dipole resonance with closure. F r o m our experience with the t2C case, below 30 M e V we use a decreasing 6 given by
6(E,)
= 1.667 X 1 0 - 2 E 2 - 6.0023 × 10 -3.
(20)
We have tried several different forms for 6(E,), and the precise form of the function does not have m o r e than an effect of a few percent on the value of (o-c}. W e are now able to evaluate Eq. (12). W e next consider a model [7] developed by Kim and Mintz for treating neutrino reactions in nuclei. This is strictly a closure approximation and should begin to be accurate at about 15 to 20 M e V about the giant dipole resonance which is assumed to saturate the matrix element. W e do not include here a full derivation which is already in the literature [7] but simply mention the salient points. As already mentioned, the starting point is Eq. (1). The quantity S+~, is defined by
dG E f~7-(ilQ~+)(q)
k)
(21)
k
with
Q(,+-)(q) - f dxJ+(x, O) e 'x'q.
(22)
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We assume that low-lying states (particularly those associated with the giant dipole resonance) saturate the sum in Eq. (21) so that we may write, applying closure,
S ~+__ - (i[O~+)(q)Q~-)(q) ]i).
(23)
Making use of translational invariance, we can show that the neutrino cross section and the total muon-capture rate depend upon essentially the same matrix element: (i] Q(+)" Q(-) ] i) + ( i l 4t~(+)o(-) [i ) . 0 40 The matrix elements themselves can be evaluated by making use of an impulse approximation for the weak current, namely A
J(~+-)(x, O) = Y'~ (F,)a~(X-ra) ,
(24)
a--1
where
(F~)a = gv6~,4 - (1 - ~ , 4 ) gA(~r,*) ~•
(25)
With these definitions we may write d/2
f -4~ [(iiQ(+).Q(-)li)+ (i[t~(+)t~(-)[i)]40 40 q2)). =Z(g~¢ + 3g 2 ) ( A1 - Z B2( A
(26)
This quantity occurs as mentioned above in the expressions for Fa-oT and for o-c. F r o m Eq. (26) the only difference is the qZ at which the matrix element is evaluated. As we have constructed a value for the quantity Fa-oa-, we may write crc as
2~rFTowEePe 1 - [ ( A - Z ) / 2 A ] 6 ( q 2) ~rc= Ci(aZ)3m~(~ 2) 1 - 3 . 1 3 [ ( A - Z ) / Z A ]
(27)
In Eq. (27), 6(q e) is given by Eq. (19). We know that Eq. (27) is valid only well above closure so we expect it to produce an underestimate for (o-c). We therefore take a worst-case (lowest-estimate) value for ~(q2), namely ~(q2) _~ 3.38 1 + 0.091-25m.
.
(28)
We now have everything necessary to calculate cross sections for the reaction Pe + -127I --> e - + X. Finally we note that from Ref. [6], we may write the cross section for inclusive neutral-current neutrino scattering as GZ(E') 2 ON=
8M2~
× 0.832D.
(29)
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25
20
15 c%(10-4%m2) 10
5
1'0
15
20
25 3~0 E (MeV)
Fig. 1. Cross section for the reaction v e -1-127I-> e calculation is via the tensor model.
+X
3'5
4'0
4'5
5'0
55
as a function of neutrino energy. This
W e are t h e r e f o r e able to evaluate the c h a r g e d - c u r r e n t cross sections in b o t h models and to find the n e u t r a l - c u r r e n t cross section.
3. Results I n Fig. 1 we plot the results of the tensor m o d e l for the c h a r g e d - c u r r e n t cross section, o-c. As already noted, at and above 30 M e V we assume 6(E~) is given by the constant, 15 MeV, but below 30 M e V we write it as given in Eq. (20). T h e r e is some arbitrariness i n h e r e n t in this but we know that 6(E~) must on average fall in this region. W e note that (o-c ) is not greatly affected by any reasonable choice of functions. F r o m Fig. 1 and the Michel s p e c t r u m we obtain the s p e c t r u m - a v e r a g e d value (o-c} = 638.52 x 1 0 - 4 2 c m 2,
(30)
w h e r e we expect a 30% error to be associated with this number. I n Fig. 2, we plot the c h a r g e d - c u r r e n t cross section using the second m o d e l described in this paper. This m o d e l yields zero below the giant dipole r e s o n a n c e at 15 M e V , as it must f r o m the assumptions used in its derivation. F r o m this result we obtain a M i c h e l - s p e c t r u m - a v e r a g e d cross section (o-c} = 303.52 X 1 0 - 4 2 c m 2.
(31)
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S.L. Mintz, M. Pourkaviani / N u c l e a r Physics A584 (1995) 665-674
12
10
crc(10-40cm2)6
01•5
20
2~5
3'0
3'5 E (MeV)
Fig. 2. Cross section for the reaction v e + 1 2 7 I - - ~ e - + X calculation is via the closure-approximation model.
4'0
4'5
5'0
55
as a function of neutrino energy. This
Finally, in Fig. 3, we plot the neutral-current results for the tensor model. This result leads to a neutral-current cross section for the reaction u + 127I--* ~, + X, averaged over the Michel spectrum, given by (o"N } = 265.18 × 10 -42 cm 2.
(32)
9 8
7 6
5 O-g( l O-4Ocrn2) 4 3 2 1
00
5
1'0
1'5
2'0
2'5
3'0
3'5
4'0
4~5
5'0
55
E (MeV) Fig. 3. Cross section for the reaction v + 127I ~ ~v-F X as a function of neutrino energy. This calculation is via the tensor model.
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W e have therefore obtained results for the inclusive charged- and the inclusive neutral-current electron-neutrino reactions in 127I. We now discuss these results in m o r e detail.
4. Discussion At present there are no published estimates of the Michel-spectrum-averaged inclusive neutrino cross sections in 127I. However, there are some informal estimates circulating which are based to a large extent on the substantial neutron excess in 127I. These estimates are of the order of 600 × 10 -42 cm 2. These are in line with our calculations as well. It is interesting to compare the small-nucleus case, i.e. 12C, with a large-nucleus case such as la7I. In the small-nucleus case, various models gave a wide range of results in this energy range. But in the large-nucleus case almost all reasonable models give a similar result. This may be due to the fact that the large nuclei with their very closely spaced levels are more like a nuclear soup so that microscopic-structure effects are not very important and a Fermi-gas t r e a t m e n t works very well. On the other hand, for a small nucleus like 12C, the structure is of p a r a m o u n t importance but is difficult to handle in calculations. This makes a semi-phenomenological treatment useful in such cases. For both of the models described here, the results are strongly tied to the total muon-capture rate. This connection is worth discussing. The function D = a 0 + bo [q21 is determined from muon-capture results as well as from additional impulse-approximation results. In the case of ~2C done earlier, b 0 was relatively small. Because I q2[ is of the order of 0.75 m ~ , D was dominated by a o. This remained true for the neutrino reaction, which was assumed to take place in the forward direction so that q2 would be quite small. Furthermore, even though the impulse-approximation value for b 0 was larger than that obtained via electronscattering data both were small enough so that the domination of D by a 0 was not affected. In the case of nuclei with large neutron excesses, this is no longer true, and because of this, a 0 is relatively smaller in these cases. But it is precisely this value for a 0 which will dominate the neutrino reaction. Thus effectively, our second model with its relatively large value for b 0 and its early cutoff at 15 MeV, the assumed center of the giant dipole resonance, yields a minimal value for ( % ) while the tensor model yields, we believe, a more realistic result. Thus we believe that a value for (o-c) of the order of (500 ~ 700) × 10 -42 cm 2 might be expected. For the region below the giant dipole resonance, i.e., the region of interest for solar-neutrino observation, we do not expect our value for o-c to be particularly accurate as we have only an impulse-approximation result to serve as a basis for the calculation. However, if it were possible to obtain inclusive electronscattering data for this (and indeed the entire range of interest), a much better calculation would be possible. Nevertheless, the result in this region, from threshold to 10 MeV, appears to favor the low end of current estimates. We should also r e m a r k that if the convergence of theoretical results is confirmed by experiment for the 127I case, it will demonstrate the ability of the G o u l a r d - P r i m a k o f f result for total muon-capture rates to work under the widest range of circumstances.
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Finally, w e wish to n o t e t h a t s o m e e a r l y results for t h e r e a c t i o n u~ + 12C ~ / x - + X have b e e n r e p o r t e d [12] f r o m Los A l a m o s . T h e m u o n n e u t r i n o s a r e p r o d u c e d f r o m i n - f l i g h t - p i o n decays a n d have an e n e r g y r a n g e f r o m 10 to 260 M e V . T h e s e p r e l i m i n a r y results a r e in g o o d a g r e e m e n t with t h e two m o d e l s d e s c r i b e d h e r e which y i e l d e s s e n t i a l l y t h e s a m e r e s u l t for t h e e n e r g y r a n g e for which t h e r e a c t i o n is p o s s i b l e (since t h e s p e c t r u m is p e a k e d well a b o v e t h e g i a n t d i p o l e r e s o n a n c e ) b u t in s t r o n g d i s a g r e e m e n t with a F e r m i - g a s - m o d e l calculation. This gives us s o m e slight a d d e d c o n f i d e n c e c o n c e r n i n g t h e s e m o d e l s . H o w e v e r , t h e r e is clearly m u c h w o r k b o t h t h e o r e t i c a l l y a n d e x p e r i m e n t a l l y to b e d o n e b e f o r e any u n d e r s t a n d i n g of t h e w e a k h a d r o n i c c u r r e n t in inclusive p r o c e s s e s c a n b e c l a i m e d .
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